Consider a one-dimensional quantum mechanical harmonic chain of N identical atoms.
This is the simplest quantum mechanical model of a lattice, and we will see how phonons arise from it. The formalism that we will develop for this model is readily generalizable to two and three dimensions. The Hamiltonian for this system is
This is the simplest quantum mechanical model of a lattice, and we will see how phonons arise from it. The formalism that we will develop for this model is readily generalizable to two and three dimensions. The Hamiltonian for this system is
where
is the mass of each atom, and 
and 
are the position and momentum operators for the 
th atom. A discussion of similar Hamiltonians may be found in the article on the quantum harmonic oscillator.We introduce a set of
"normal coordinates" 
, defined as the discrete Fourier transforms of the 
's and 
"conjugate momenta" 
defined as the Fourier transforms of the 
's:
The quantity
will turn out to be the wave number of the phonon, i.e. 
divided by the wavelength. It takes on quantized values, because the number of atoms is finite.The form of the quantization depends on the choice of boundary conditions; for simplicity, we impose periodic boundary conditions, defining the
th atom as equivalent to the first atom. Physically, this corresponds to joining the chain at its ends. The resulting quantization is
The upper bound to
comes from the minimum wavelength imposed by the lattice spacing 
, as discussed above.By inverting the discrete Fourier transforms to express the
's in terms of the 
's and the 
's in terms of the 
's, and using the canonical commutation relations between the 
's and 
's, we can show that
In other words, the normal coordinates and their conjugate momenta obey the same commutation relations as position and momentum operators! Writing the Hamiltonian in terms of these quantities,
where
Notice that the couplings between the position variables have been transformed away; if the
's and 
's were Hermitian (which they are not), the transformed Hamiltonian would describe 
uncoupled harmonic oscillators.This may be generalized to a three-dimensional lattice. The wave number k is replaced by a three-dimensional wave vector k. Furthermore, each k is now associated with three normal coordinates.
The new indices s = 1, 2, 3 label the polarization[disambiguation needed] of the phonons. In the one dimensional model, the atoms were restricted to moving along the line, so the phonons corresponded to longitudinal waves. In three dimensions, vibration is not restricted to the direction of propagation, and can also occur in the perpendicular planes, like transverse waves. This gives rise to the additional normal coordinates, which, as the form of the Hamiltonian indicates, we may view as independent species of phonons.
Kevin M Contreras H
Electrónica del Estado Sólido
http://en.wikipedia.org/wiki/Phonon
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